Predicate Transformers, (co)Monads and Resolutions

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Pierre Hyvernat

To cite this version:

Pierre Hyvernat. Predicate Transformers, (co)Monads and Resolutions. 2004. �hal-00395633�

Predicate Transformers, (co)Monads and Resolutions

Pierre Hyvernat

1

Institut math´ematique de Luminy, Marseille, France

2

Chalmers Institute of Technology, G¨ oteborg, Sweden hyvernat@iml.univ-mrs.fr

Abstract. This short note contains random thoughts about a factoriza- tion theorem for closure/interior operators on a powerset which is remi- niscent to the notion of resolution for a monad/comonad. The question originated from formal topology but is interesting in itself.

The result holds constructively (even if it classically has several varia- tions); but usually not predicatively (in the sense that the interpolant will no be given by a set). For those not familiar with predicativity is- sues, we look at a “classical” version where we bound the size of the interpolant.

Introduction

A very general theorem states that any monotonic operator F : P(X ) → P(Y ) can be factorized in the form P (X) → P (Z ) → P(Y ), where Z is an appropriate set; and the first predicate transformer commutes with arbitrary unions and the second commutes with arbitrary intersections.

We prove similar factorization for interior and closure operators on a power- set; the idea being to “resolve” the operator as is usually done for (co)monad in categories. We then look at the constructive version of those factorizations.

1 Relations and Predicate Transformers

We start by introducing the basic notions:

Definition 1. If X and Y are sets, a (binary) relation between X and Y is a subset of the cartesian product X × Y . The converse of a relation r ⊆ X × Y is the relation r

⊆ Y × X defined as (y, x) ∈ r

≡ (x, y) ∈ r.

A predicate transformer from X to Y is an operator from the powerset P (X ) to the powerset P (Y ).

Since most of our predicate transformers will be monotonic (with respect to

inclusion), we drop the adjective when no confusion is possible.

Definition 2. Suppose r is a relation between X and Y ; we define two mono- tonic predicate transformers from Y to X :

hri : P(Y ) → P (X)

V 7→ {x ∈ X | (∃y ∈ Y ) (x, y) ∈ r & y ∈ V } and

[r] : P(Y ) → P (X)

V 7→ {x ∈ X | (∀y ∈ Y ) (x, y) ∈ r ⇒ y ∈ V } ; and an antitonic predicate transformer:

⌊r⌉ : P(Y ) → P(X )

V 7→ {x ∈ X | (∀y ∈ V ) (x, y) ∈ r} . Concerning notation:

– hri and [r] are somewhat common in the refinement calculus, even though the main reference ([1]) uses {r} instead of hri. The problem is that this clashes with set theoretic notation.

– In [2], Birkhoff uses V

for ⌊r⌉, but this supposes that r is clear from the context. (The notation V

would then be ⌊r

⌉(U ).)

– The linear logic community would use V

for the same thing, but this also supposes that the relation is called “⊥”.

We will always be in a “typed” context; i.e. subsets will always be subset of some ambient set. We write ¬ for complementation with respect to that ambient set.

Lemma 1. Suppose r is a relation between X and Y ; we have:

1. hri · ¬ = ¬ · [r];

2. h¬ri = ¬ · ⌊r⌉. (where (x, y) ∈ ¬r ≡ (x, y) ∈ / r) A very interesting property is the following Galois connections:

Lemma 2. Suppose r is a relation between X and Y ; then hri ⊢ [r], and ⌊r⌉ is Galois-connected to itself:

1. hri(V ) ⊆ U ⇔ V ⊆ [r

](U );

2. U ⊆ ⌊r⌉(V ) ⇔ V ⊆ ⌊r

⌉(U ).

Those predicate transformers satisfy:

Lemma 3. If r is a relation between X and Y , then

– hri commutes with arbitrary unions; (i.e. it is a sup-lattice morphism

) – [r] commutes with arbitrary intersections; (i.e. it is an inf-lattice morphism) – ⌊r⌉ transforms arbitrary unions into intersections.

3

All of our lattices are complete, so we do not bother writing “complete” all the

time...

3

and moreover:

– any sup-lattice morphism from P (Y ) to P(X ) is of the form hri for some r ⊆ X × Y ;

– any inf-lattice morphism from P (Y ) to P(Y ) is of the form [r] for some r ⊆ X × Y ;

– any predicate transformer from Y to X taking arbitrary unions to intersec- tions is of the form ⌊r⌉ for some r ⊆ X × Y .

Proof. The sup-lattice part is easy; and the rest is an application of Lemma 1.

⊓

⊔ Just like it is possible to factorize any relation as the composition of a total function and the inverse of a total function, it is possible to factorize an mono- tonic predicate transformer as the composition of a hri and a [s] (see [3] for a detailed categorical construction).

Proposition 1. Suppose F is a monotonic predicate transformer from X to Y ; then there is an X

and there are relations s ⊆ X

× X and r ⊆ Y × X

such that F = hri · [s].

Proof. Let F be a monotonic predicate transformer, and define X

= P (X ) together with (U, x) ∈ s ≡ x ∈ U and (y, U) ∈ r ≡ y ∈ F (U ). We have:

y ∈ hri · [s](U )

⇔ { definition of hri }

(∃V ∈ X

) (y, V ) ∈ r & V ∈ [s](U )

⇔ { definition of X

and r } (∃V ⊆ X) y ∈ F (V ) & V ∈ [s](U )

⇔ { definition of [s] }

(∃V ⊆ X) y ∈ F (V ) & (∀x) (V, x) ∈ s ⇒ x ∈ U

⇔ { definition of s }

(∃V ⊆ X) y ∈ F (V ) & (∀x) V ⊆ U

⇔ { since F is monotonic } y ∈ F(U )

The result thus holds, but the proof doesn’t bring much information... ⊓ ⊔ And as a direct application of Lemma 1:

Corollary 1. Any monotonic predicate transformer can be factorized as a [r]·hsi or as a ⌊r⌉ · ⌊s⌉.

Similarly, any antitonic predicate transformer can be written has one of ⌊r⌉ · [s],

⌊r⌉ · hsi, hri · ⌊s⌉ or [r] · ⌊s⌉.

2 Interior and Closure Operators

Definition 3. If (X, ≤) is a partial order, we say that F : X → X is an interior operator if:

– F is monotonic;

– F is contractive: F (x) ≤ x;

– F(x) ≤ F F (x).

We say that it is a closure operator if:

– F is monotonic;

– F is expansive: x ≤ F (x);

– F F (x) ≤ F (x).

It is well known that the composition of two Galois connected operators yield interior/closure operators, so that we have:

Lemma 4. If r is a relation between X and Y , then – hri · [r

] is an interior operator on P(X );

– [r] · hr

i is a closure operator on P (X );

– ⌊r⌉ · ⌊r

⌉ is a closure operator on P (X).

Some other consequences of the Galois connection are listed below:

– hri · [r

] · hri = hri;

– [r] · hr

i · [r] = [r];

– ⌊r⌉ · ⌊r

⌉ · ⌊r⌉ = ⌊r⌉.

The problem is now to mimic Proposition 1.

Definition 4. If F is an interior operator on P (X ), a resolution for F is given by a set Y (called the interpolant) together with a relation r ⊆ Y × X such that F = hri · [r

].

Proposition 2. If F is an interior operator on P (X ), then it has a resolution.

The proof relies on the following lemma:

Lemma 5. Let F be an interior operator on a complete sup-lattice (X, ≤, W );

write Fix (F) for the collection of fixed-point for F . We have that ( Fix (F ), ≤, W ) is a complete sup-lattice; and for any x ∈ X

F(x) = _

y ∈ Fix(F) | y ≤ x . Proof. That (Fix(F), ≤, W

) is a complete sup-lattice is left as an easy exercise;

for the second point, let x ∈ X;

– we know that F (x) is a fixed point of F, and that F (x) ≤ x. This implies

that F(x) ∈ {y ∈ Fix(F) | y ≤ x}; and so F (x) ≤ W {y ∈ Fix(F ) | y ≤ x};

5 – suppose y ∈ Fix(F ) and y ≤ x; this implies that F (y) ≤ F (x), i.e. that

y ≤ F (x). We can conclude that W {y ∈ Fix(F) | y ≤ x} ≤ F(x).

⊓

⊔ Proof (of proposition 2). Suppose F is an interior operator on P(X ); define Y = Fix(F ) and (U, x) ∈ r ≡ x ∈ U . We have:

x ∈ hri · [r

](U )

⇔ { definition of r }

∃V ∈ Fix(F )

x ∈ V & (∀y) y ∈ V ⇒ y ∈ U

⇔

∃V ∈ Fix(F )

x ∈ V & V ⊆ U

⇔

x ∈ S {V ∈ Fix(F ) | V ⊆ U }

⇔ { Lemma 5 } x ∈ F (U )

which concludes the proof. ⊓ ⊔

Just like for Proposition 1, the statement of the theorem is interesting, but the proof hardly tells us anything about the structure of F . To gain a little more information about F , we will try to “bound” the size of the interpolant set Y . Definition 5. If (X, ≤, W

) is a complete sup-lattice, we say that a family (x

)

of element of X is a basis if, for any y ∈ X , we have y = _

{x

| x

≤ y} .

Corollary 2. Suppose F is an interior operator on P (X ); if Fix (F ), ⊆, S has a basis of cardinality κ, then we can find a resolution of F with an interpolant Y of cardinality κ.

Proof. It is easy to see that in the above proof of Proposition 2, we can replace

Fix(F) by a basis of Fix(F ), ⊆, S . ⊓ ⊔

In particular, if there is a basis of Fix (F) which has cardinality less that the cardinality of X ; we can use X as the interpolant and use a relation r ⊆ X × X to obtain a resolution of F .

We now show that this result is optimal:

Lemma 6. Let F be an interior operator on P(X ); and suppose there are no basis of Fix(F) of cardinality κ; then there is no interpolant of cardinality less than κ.

Proof. To show that, we will construct a basis of Fix(F ) indexed by any inter-

polant for F. Suppose Y and r form a resolution for F . For any y ∈ Y , define

U

≡ hri{y}. We will show that (U

)

is a basis for Fix(F ).

Each U

is a fixed point for F :

U

= hri{y} { definition }

= hri · [r

] · hri{y} { second part of Lemma 4 }

= F · hri{y} { r is a resolution of F }

= F (U

) { definition } Let U be a fixed point of F:

U = hri · [r

](U ) { U is a fixed point of F }

= hri S {y} | y ∈ [r

](U ) { basic logic }

= S hri{y} | {y} ⊆ [r

](U ) { hri commutes with unions }

= S hri{y} | hri{y} ⊆ U { Galois connection between hri and [r

] }

= S {U

| U

⊆ U } { definition }

which concludes the proof that (U

)

is a basis for Fix(F). ⊓ ⊔ Let’s look at an example of interior operator on P (X) which cannot be resolved using X as an interpolant. Let X be a countable infinite set (natural numbers for example); and define F : P (X) → P (X) as follows:

F(U ) =

∅ if U is finite U if U is infinite

The sup-lattice Fix(F ) is given by the collection of infinite subsets of X ; and this lattice doesn’t have a countable basis. To prove that, it is enough to do it for any particular countable infinite set. Take C to be the set of finite strings over {0, 1}. If α is an infinite string of 0’s and 1’s, define U

⊆ C to be the set of finite prefixes of α. Each U

is an element of Fix(F ); but no countable family of infinite subsets can “generate” all the U

: since α 6= β implies that U

∩ U

is finite, if V

⊆ U

and V

⊆ U

then i 6= j. In other words, a family which generates all the U

’s needs to have the cardinality of the collection of the U

’s, i.e. uncountable.

Using Lemma 1, we can now extend all what has been done for interior operators for closure operators: if F is a closure operator on P(X ), then ¬ · F · ¬ is an interior operator on P(X ).

Corollary 3. If F is a closure operator on P (X ), then it has a resolution as a composition [r] · hr

i or as ⌊r⌉ · ⌊r

⌉.

As for interior operators, the possible cardinalities of the interpolant are given by the cardinalities of the bases for the inf-lattice Fix(F ).

In particular, for linear logicians, it is not the case that any closure operator can be written as a biorthogonal...

3 Comonad and Monads

It seems that the traditional way to look at monads in a category is to see them as a kind of generalized monoid; at least in my part of the world. Another view

4

which has given me a much better understanding of what (co)monads are

7 is to view them as a generalization of closure operators. This is the view taken in the introduction of [4].

Definition 6. A monad on a category C is a morphism F : C → C together with two natural transformations η : Id

→ F and µ : F F → F s.t. some diagram commute.

If one takes the partial order category P (X ), then a monad is an operator on P (X) s.t.:

– it acts on morphisms: if i : U ⊆ V then F

: F (U ) ⊆ F (V ); i.e. F is monotonic;

– there is a natural transformation η

: U ⊆ F (U );

– there is a natural transformation µ

: F F (U ) ⊆ F(U ).

All the “coherence” conditions are trivially satisfied in a partial order category, since every diagram commute! What we’ve just shown is that a monad on P (X ) is nothing more than a closure operator on P(X ); and vice and versa. Similarly, a comonad corresponds to an interior operator.

In categories, a resolution for a monad corresponds to factorizing the func- tor F as the composition of two adjoint functors. Adjointness H ⊢ G between the functors G : D → C and H : C → D in a locally small category means that C[A, G(B)] ≃ D[H (A), B] which, in the case of a partial order category sim- plifies to “U ⊆ G(V ) iff H (U ) ⊆ V ” i.e. is exactly the Galois connection H ⊢ G.

Category theory tells us that there always is a resolution since we have two degenerate resolutions: something like F = Id· F and F = F ·Id. The first one is given by the Eleinberg-Moore category. In the context of order theory, it amounts to taking: interpolant category to partial orders: if F is a closure operator on X , take E(X, F ) to be the the collection of fixed points for F , with the ordering inherited from X .

We have F : X → E (X, F ) and Id : E(X, F ) → X which are adjoint: F ⊢ Id.

x ≤ Id(f )

⇒ { F is monotonic } F(x) ≤ F (f )

⇔ { f ∈ E (X, F ), i.e. f is a fixed point for F } F(x) ≤ f

and F (x) ≤ f ⇒ x ≤ f = Id(f ) since x ≤ F (x).

The second resolution (F · Id ) is obtained via the Kleisli categry. In the context of order theory, it amounts to taking the order (X, ⊑) where x ⊑ y iff F (x) ≤ F(y).

What is important to us is that those resolutions are trivial and uninteresting.

More abstract nonsense states that there is a category of resolutions for any given monad; and that those two trivial resolutions are respectively initial/terminal.

5

In a partial order, an algebra for F is just a post fixed point: x ≤ F (x). If F is a

closure, then it is also a fixed point for F .

What we have done with Proposition 2 and Corollary 3 corresponds to finding a non-trivial resolution for the monad corresponding to the interior/closure oper- ator. It is even more than a resolution, since the interpolant is itself a complete and cocomplete category (and one of the functors preserves limits while the other one preserves colimits; but this is a general fact about adjoints). The feeling is that this resolution lies “exactly in the middle” between the initial and terminal resolutions. I don’t know if this kind of “strong” resolution has been considered in category theory.

The resolution constructed here is quite different from the Eleinberg-Moore resolution (even though it uses the fixed-points as a basis). We do not construct functors from P (X) to E(X, F ) and back; but from P(X ) to P E(X, F )

and back. In particular, as noted above, the interpolant is complete and cocomplete;

which is not the case for the Eleinberg-Moore category: fixed points for an in- terior are closed under unions but not under intersections; and conversely for closure operators.

4 Revisiting Section 2 in a Constructive Setting

4.1 Impredicative

In the previous section we used Lemma 1 to generalize results on interior to closures. It is thus natural to ask whether (1) we can make the original proof constructive; (2) we can avoid using this lemma and prove the result for closure constructively. I will not into the details but just provide some hints about that.

– Galois connections from Lemma 2 are constructive.

– Lemma 3: the first part is trivial. The second part is easy: if F commutes with unions, take (x, y) ∈ r iff y ∈ F {x}; if F commutes with intersections, take (x, y) ∈ r iff (∀U ) y ∈ F (U ) ⇒ x ∈ U ; and if F transforms unions into intersections, take (x, y) ∈ r iff y ∈ F {x}.

– Proposition 1: the proof is constructive.

– Lemma 3 is constructive.

– Lemma 5 and the corresponding lemma for closure operator are constructive.

– Proposition 2 is constructive.

– we can mimic the proof of Proposition 2 to obtain a resolution of a closure operator as F = ⌊r⌉ · ⌊r

⌉, but not to obtain a resolution as F = [r] · hr

i.

– I doubt we can constructively obtain a resolution of a closure operator as F = [r] · hr

i.

– all of the lemmas about the size of interpolant are constructive at least if we read then as “if B is a basis for Fix(F) then we can use B as an interpolant”.

As a proof of concept, all this (except the last point) has been proved in the proof assistant COQ.

6

i.e. if C is a complete and cocomplete category and F is a monad on C, a “strong resolution” is a resolution with a complete and cocomplete interpolant. Any reference to something similar in the literature would be most welcome.

7

i.e. is a complete lattice since we deal with partial orders

8

proof scripts available from http://iml.univ-mrs.fr/~hyvernat/academics.html

9

4.2 Predicative

In a predicative setting like Martin-L¨ of type theory (see [5]) or CZF (constructive ZF, see [6]) set theory, many of the results are not provable. The reason being that we do not allow quantification on a power-set. The main result about resolution would become something like:

Proposition 3. Suppose Fix(F) (proper type) has a set-indexed basis, then F as a resolution as hri·[r

] (if F is an interior) or as ⌊r⌉·⌊r

⌉ (if F is a closure).

If F as a resolution, then Fix(F ) as a set-index basis.

Note that having a set-indexed basis is equivalent to being “set presented” in the terminology of P. Aczel ([6,7]). Note that in the case of an interior operator, this implies that the predicate transformer is “set-based” (in the sense that it has a factorization as in Proposition 1, where the interpolant X

is a set). It doesn’t seem that the existence of a resolution for a closure operator implies that the original predicate transformer is itself set-presented.

Conclusion

Nothing revolutionary has really been done, but the statement of Propositions 1 and 2 is, in an abstract setting, quite neat. However, as the proofs show, this is mostly abstract nonsense. The best example is probably Proposition 1, where y ∈ F (U ) is factorized as “there is a V such that s ∈ F(V ) and V ⊆ U ”. The proof of Proposition 2 is slightly subtler, but is hardly interesting. In the end, the most interesting and informative thing is probably Lemma 6, in its “positive”

version: if F has Y as an interpolant, then Fix(F) has a basis indexed by Y , which is hardly a breakthrough in mathematics...

I do nevertheless hope that it might interest some people, since while Propo- sition 1 is known to many (especially the refinement calculus people), it seems that Proposition 2 isn’t stated anywhere. I also hope the link between monad and closure operator (together with the Eleinberg-Moore category being the partial order of fixed points) will gain in popularity, as I see it as a much better way of seeing monads, at least as far as intuition is concerned.

A final word about the motivation for this: the starting point was the question about whether it is possible to represent any “basic topology” (see the forth- coming [8]) as the formal side of a basic pair, impredicatively speaking. A basic topology is a structure (X, A, J ) where X is a set, and A and J are closure and interior operators on P (X ) such that A(U ))(J (V ) ⇒ U )((J (V ).

The answer is obviously no since the A and J arising from a formal pair are classically dual (i.e. A · ¬ = ¬ · J ) and there are basic topologies which are provably not dual.

The question then turned into: “can any interior operator be written as the for- mal interior of a basic pair?” and similar for closure operators. The answers are yes (Proposition 2) and I don’t know (the constructive resolution of a closure is of the form ⌊r⌉ · ⌊r

⌉, and not of the form [r] · hr

i.)

9

U ))(V is the constructive version of U ∩ V 6= ∅.

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